Defining Limits and Using Limit Notation AB & BC

The statement $\lim_{x\to a} f(x) = L$ says the outputs can be forced arbitrarily close to $L$ by taking inputs close enough to $a$, with $x = a$ itself deliberately excluded. So the limit is independent of $f(a)$: it can exist where the function is undefined, and it can differ from a function value that is defined. One-sided notation, $x\to a^-$ and $x\to a^+$, records which direction the inputs come from, and the two-sided limit exists exactly when both one-sided limits exist and agree.

Two errors dominate. The first is reading the limit off the function value, so a hole, a jump, or a piecewise redefinition at the point gets ignored and the plotted dot is reported instead of the height the curve approaches. The second is taking a two-sided limit at a join without checking both sides, which shows up at piecewise boundaries, at absolute-value corners such as $\frac{|x|}{x}$ at 0, and at vertical asymptotes where the sides run to opposite infinities.